Finite Group Databasebeta

← #69153

Subgroups — #69153

Order 384.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5679C211yesmaximal, characteristic, complemented
2#5061C211yesmaximal, characteristic, complemented
3#4742C211yesmaximal, characteristic, complemented
4#4741C211yesmaximal, characteristic
5C3×#354C211yesmaximal, characteristic, complemented
6#4472C211yesmaximal, characteristic, complemented
7#4471C211yesmaximal, characteristic, complemented
8#2197—31nomaximal, Sylow
9#1004C2211yescharacteristic
10C3×#124C2211yescharacteristic
11#946—21no—
12#942—21no—
13#937C2211yescharacteristic
14#935C2211yescharacteristic
15#886—21no—
16#883—21no—
17#882—21no—
18#67⋊C4—21no—
19D3×C22⋊C4—21no—
20#879—21no—
21#872—21no—
22C3×#99C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#833C2211yescharacteristic
25#806—21no—
26#806—21no—
27#805—21no—
28#551—31no—
29#458—31no—
30B(4,𝔽2)—31no—
31#455—31no—
32#354D311yescharacteristic, complemented
33#352—31no—
34AΣL(1,𝔽16)—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic, complemented
37C2×C3×D4C2311yescharacteristic
38C2×#67—21no—
39C2×#67—21no—
40C2×#67—21no—
41C22×Dic3D411yescharacteristic, complemented
42#237—41no—
43D3×D4—41no—
44#235—21no—
45C2×D12—41no—
46C2×C4×D3—21no—
47C2×C4×D3—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C22⋊C4—21no—
51C3×C22⋊C4—21no—
52C3×C42D411yescharacteristic, complemented
53#217—21no—
54#217D411yescharacteristic
55#217D411yescharacteristic, complemented
56#216—41no—
57#215—41no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—41no—
62(C2×D3)⋊C4—41no—
63C4⋊Dic3—21no—
64C4⋊Dic3—41no—
65Dic3⋊C4—21no—
66Dic3⋊C4—21no—
67Dic3⋊C4—21no—
68C4×Dic3—21no—
69#208—21no—
70H(5,𝔽2)—31no—
71#137—61no—
72#126—61no—
73#124C2×D311yescharacteristic
74#123—31no—
75#123—61no—
76#123—61no—
77C22⋊Q8—61no—
78#120—31no—
79#120—61no—
80C4×D4—61no—
81#117—61no—
82C2×C22⋊C4—61no—
83#104—61no—
84#100—31no—
85#99—61no—
86#99—61no—
87#99C2×D311yescharacteristic
88#99C2×D311yescharacteristic
89C23×C3C2×D411yescharacteristic
90C23×C3C2×D411yescharacteristic
91C22×D3—21no—
92C22×D3—21no—
93C22×D3—21no—
94C22×D3—42no—
95C22×D3—41no—
96C3×Q8—41no—
97C3×D4—41no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C2×D411yescharacteristic
103#67—42no—
104#67—41no—
105#67—41no—
106#67—41no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—41no—
113D12—21no—
114D12—82no—
115C4×D3—41no—
116C4×D3—42no—
117C4×D3—81no—
118Dic6—21no—
119C3⋊C8—41no—
120C24—31no—
121#41—61no—
122#41—121no—
123C2×Q8D1211yescharacteristic, complemented
124C2×D4—61no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4—121no—
128C2×D4—121no—
129C2×D4C22×D311yescharacteristic
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133Q16—121no—
134SD16—121no—
135M16—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—121no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—31no—
146C22⋊C4—31no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—121no—
152C22⋊C4—121no—
153C42—61no—
154C42D1211yescharacteristic, complemented
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3#12011yescharacteristic
160C2×D3—22no—
161C2×D3—41no—
162C2×D3—41no—
163C2×D3—42no—
164C2×D3—41no—
165C2×D3—81no—
166C2×D3—81no—
167C2×D3—81no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—81no—
172C3×C4—81no—
173Dic3—41no—
174Dic3—41no—
175Dic3—41no—
176Dic3—41no—
177C23—61no—
178C23—61no—
179C23—61no—
180C23—121no—
181C23—122no—
182C23D3×D411yescharacteristic
183C23D3×D411yescharacteristic
184Q8—41no—
185Q8—61no—
186D4—41no—
187D4—61no—
188D4—121no—
189D4—121no—
190D4—122no—
191D4—121no—
192D4—242no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—121no—
203C2×C4—122no—
204C2×C4—121no—
205C2×C4—241no—
206C2×C4C2×D1211yescharacteristic
207C8—121no—
208C2×C3—21no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3B(4,𝔽2)11yescharacteristic
212D3—42no—
213D3—81no—
214C22—21no—
215C22—21no—
216C22—41no—
217C22—41no—
218C22—62no—
219C22—121no—
220C22—122no—
221C22—121no—
222C22—121no—
223C22—241no—
224C22—241no—
225C22—241no—
226C22#88211yescharacteristic
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—81no—
232C4—121no—
233C4—121no—
234C4—121no—
235C4—121no—
236C3#219711yescharacteristic, complemented, Sylow
237C2—21no—
238C2—41no—
239C2—41no—
240C2—122no—
241C2—241no—
242C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×#354C2
socleC2×C3B(4,𝔽2)
prime radical#4741C2
hypercenter#354D3
hypocenterC3#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #5679C2semidirect
2. #5061C2semidirect
3. #4742C2semidirect
5. C3×#354C2semidirect
6. #4472C2semidirect
7. #4471C2semidirect
32. #354D3semidirect
36. C2×C3×Q8D4semidirect
41. C22×Dic3D4semidirect
52. C3×C42D4semidirect
55. #217D4semidirect
123. C2×Q8D12semidirect
154. C42D12semidirect
236. C3#2197semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#56792
#50612
#47422
#47412
C3×#3542
#44722
#44712
#21973